Hence \(\lim \limits _{n\rightarrow \infty }\dfrac{1}{n}\sum _{i=1}^{\infty }\mathbb {E}\lbrace \psi _i(\mathbf {Y}_i,\mathbf {\theta })\rbrace =0\)  implies \( \mathbf {\theta }=\mathbf {\theta }_0\) . Since \(\widehat{{\mathbf {\theta }}}_{mix\cdot em}\)  is always \(O_p(1)\) , by Proposition 5.5 in [1]}, \(\widehat{{\mathbf {\theta }}}_{mix\cdot em} \overset{p}{\rightarrow }\mathbf {\theta }_0\) .
